\(\eta \)  is an irregular boundary point. By Kellogg's Theorem, cf. [1]}, \(P = \partial f(X) \cap \mathbb {D}\)  is a relatively closed polar set in \(\mathbb {D}\) . Suppose \(z_0 \in \mathbb {D} \setminus \overline{f(X)}\) . Then for some \(\epsilon > 0\) , \(f(X) \subset \mathbb {D} \setminus \overline{\mathbb {D}(z_0, \epsilon )}\) . Let \(k\)  be the harmonic function defined on the latter set with Dirichlet boundary data
\(k(z) ={\left\lbrace \begin{array}{ll}0, & z \in \partial \mathbb {D} \\-\log (|z_0| + \epsilon ), & z \in \partial \mathbb {D}(z_0, \epsilon ).\end{array}\right.}\) 
